Question

Show that the points A(1, 2, 3), B(-1, -2, -1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram. Show that ABCD is not a rectangle.

Answer 1

To prove: Points A, B, C, D form parallelogram. 

Formula: The distance between two points (x₁,y₁,z₁) and (x₂,y₂,z₂) is given by

D = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

Here, 

(x₁,y₁,z₁)= (1, 2, 3) 

(x₂,y₂,z₂)= (-1, -2, -1) 

(x₃,y₃,z₃)= (2, 3, 2) 

(x₄,y₄,z₄)= (4, 7, 6)

Length AB = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

Length CD = \(\sqrt{(x_4-x_3)^2+(y_4-y_3)^2+(z_4-z_3)^2}\)

Length AC = \(\sqrt{(x_3-x_1)^2+(y_3-y_1)^2+(z_3-z_1)^2}\)

Here, AB = CD which are opposite sides of polygon. 

BC = AD which are opposite sides of polygon. 

Also the diagonals AC and BD are not equal in length. 

Hence, the polygon is not a rectangle.